Description

$\mathbf{\sigma}^1$ is a non-linear viscous stress, $\mathbf{\sigma}^2$ is a non-linear
elastic stress and $\mathbf{\sigma}^3$ is a linear elastic stress component.
The rheological model is depicted below.

where $f$ is a user defined FUNCTION or CURVE with ID cid.
FUNCTION allows the viscosity to depend on both the effective geometric strain and the strain rate.
If using a CURVE, the viscosity is a function of $\dot{\bar{\mathbf{\epsilon}}}$ only.
$\dot{\bar{\mathbf{\epsilon}}}$ is a smeared out strain rate measure:

Note that $\dot{\bar{\mathbf{\epsilon}}} = \dot{\mathbf{\epsilon}}$ if $c_{dec}=0$.
The non-linear elastic stress $\mathbf{\sigma}^2$ is defined to grow quadratically with the total deviatoric
strain $\mathbf{\epsilon}_{dev}$:

$p$ is the hydrostatic pressure and $\mathbf{\epsilon}_{dev}^e$ is the deviatoric part of the linear elastic strain tensor $\mathbf{\epsilon}^e$
in the relation:

$\mathbf{\epsilon} = \mathbf{\epsilon}^e + \mathbf{\epsilon}^p$

where $\mathbf{\epsilon}^p$ is a plastic strain tensor. The plasticity model is based on a von Mises
effective stress definition and an iso-choric plastic flow law. The plastic flow stress is defined as:

where $K$ is the linear bulk modulus, $\epsilon_v$ is the volumetric strain.
$\alpha_T$ is the thermal expansion coefficient and $T_{ref}$ is the reference temperature (see PROP_THERMAL).

Example

Gelatin like material

This is a complete model of a rigid sphere impacting a cylinder of a gelatin like visco-elastic material. Plasticity is turned off
by setting the initial yield stress to a very large value. The FUNCTION with ID 20 defines the dynamic viscosity
as a function of the total effective strain rate "rate" and the total geometrical strain "egeo".